Spatio-Temporal Disease Mapping: Malaria in West Africa
STAT 647 Fall 2024
Wesley Halstead and Jack Kissell
Goals
- Analyze the spatio-temporal structure of Malaria infection in Nigeria
- Determine which covariates are likely correlated with infection rates
- Compare model frameworks to determine the best level of complexity for the data
Background (WHO 2024)
- “Malaria is a life-threatening disease spread to humans by some types of mosquitoes.”
- “It is mostly found in tropical countries, [but] it is preventable and curable.”
- “Globally in 2022, there were an estimated 249 million malaria cases and 608 000 malaria deaths in 85 countries.”
- “[The African continent] carries a disproportionately high share of the global malaria burden.”
- Many countries have been able to eliminate the disease entirely
Previous Analyses
- Previous attempts at spatiotemporal mapping of Malaria data have been done in the past.
- Ogunsakin et al.’s paper GIS-based spatiotemporal mapping of malaria prevalence and exploration of environmental inequalities models on clusters of point data.
- Identified Nigeria as the country with the highest Malaria incidence
- Showed importance of climate features in modeling
- Data mostly in southern areas of Nigeria
Data Collection
- Malaria Infection rates in Nigeria collected from Malaria Atlas Project
- Measured at the state level between 2010 and 2022
- Covariate climate data collected from World Bank Climate Change Knowledge Portal
- We include climate metrics as a predictor since previous analyses have shown them to be statistically significant
- State boundary shapefiles collected from World Bank Data Catalog
Spatial Exploration
- Plotting incidence rate by location shows groups of high and low infection regions
- Suggests need for a spatial effect
Temporal Analysis
General Modeling Framework
We employ a variation of the model described by Lance Waller and Brad Carlin in Chapter 14 of the Handbook of Spatial Statistics
Bayesian approach in which spatio-temporal structure is modeled via sets of autocorrelated random effects \[
Y_{kt} \sim \text{Poisson}(\mu_{kt}) \\
\ln(\mu_{kt}) = \beta x^T_k + O_{kt} + \psi_{kt} \\
O_{kt} \text{ is some offset term} \\
\psi_{kt} \text{ is some spatio-temporal random effect}
\]
We plan on modeling \(\psi_{kt}\) with a CAR prior as porposed by Knorr-Held (2000)
Implemented using R package CARBayes and CARBayesST
Knorr-Held CAR prior
\[\begin{align}
\psi_{kt} &= \phi_k + \delta_t \\
\phi_k | \phi_{-k} \mathbf{W} &\sim N(\frac{\rho_S \sum_{j=1}^k w_{kj} \phi_j}{\rho_S \sum_{j=1}^k w_{kj} +1 - \rho_S} , \frac{\tau_S^2}{\rho_S \sum_{j=1}^k w_{kj} +1 - \rho_S }) \\
\delta_k | \delta_{-k} \mathbf{D} &\sim N(\frac{\rho_T \sum_{j=1}^N d_{kj} \delta_j}{\rho_T \sum_{j=1}^N d_{kj} +1 - \rho_T} , \frac{\tau_T^2}{\rho_T \sum_{j=1}^k d_{tj} +1 - \rho_T }) \\
\tau^2_S, \tau^2_T &\sim \text{Inverse-Gamma}(a,b) \\
\rho_S, \rho_T &\sim \text{Uniform}(0,1)
\end{align}\]
- \(w_{ij}\) is \(1\) if region \(i\) and region \(j\) share a boundary. Zero otherwise.
- \(\delta_{ij}\) is \(1\) if time \(i\) comes immediately before or immediately after time \(j\). Zero otherwise.
Expected Counts as Offset Term
- 12 years of measurements across the 37 states in Nigeria
- Responses, \(Y_{kt}\) are the number of newly diagnosed cases of Malaria (plasmodium falciparum) in state \(k\) during year \(t\).
- We calculate the expected number of cases in state \(k\) during year \(t\), \(E_{kt}\), as \[E_{kt} = P_{kt}(\frac{\sum_{k}\sum_{t} Y_{kt}}{\sum_{k}\sum_{t} P_{kt}})\]
- Where \(P_{kt}\) is the population of state \(k\) in year \(t\).
- That is, we expect a constant rate across all states and years
- The log of this term will be used as our offset in our model
- Thus \(\mu_{kt} = E_{kt}\exp({\beta x^T_k + \psi_{kt}})\)
- The term \(\exp({\beta x^T_k + \psi_{kt}})\) is known as the relative risk
Covariates
- Climate covariates included:
- Mean of the average monthly temperature
- Mean of the average monthly minimum temperature
- Mean of the average monthly precipitation for each state
- Maximum temperature was not included due to collinearity with average monthly temperature
- Also include population
- Covariates are scaled using R’s
scale() function
Fixed Effects Model
- We start by fitting a fixed effects model (ignoring spatio-temporal effects)
![]()
Fixed Effects Model
Using means as Bayes estimators of \(\beta\), residuals of fixed effects model still show some spatio-temporal effects
Spatial Model
- Model with CAR modeled random spatial effect (no temporal effect)
- Lower DIC suggests much better fit
![]()
Spatial Model
Mean of Rho converges around \(0.77\) suggesting strong spatial correlation. ![]()
Spatio-temporal Model
- Model with CAR modeled random spatial and temporal effect
- Increased DIC value suggests any improvements in fit do not outweigh penalizations for extra parameters
- Possibly overfitting to our data
![]()
Spatio-temporal Model
- Relatively high values for both correlation estimates suggest strong correlation of values with spatial or temporal closeness
![]()
Results
- Non-zero credible intervals for all of our coefficient estimates
- Strong evidence for benefits of spatial modeling
- Benefits seen for spatio-temporal, but more work needs to be done to see if we are over-parameterizing our model
Moving Forward
- Include more covariates in the model
- Possibly socio-demographic factors
- Latitude
- Experiment with different parameterizations or CAR priors
- Regional analysis by including more countries